Mathematical methods and techniques in signal processing [E9 252 (3:0) – Spring 2024]

Instructor:  
Shayan S. Garani

Video lectures available at NPTEL and YouTube.

Pre-requisities:

Digital signal processing at the undergrad level.

Course Syllabus:

  • Review of basic signals, systems and signal space: Review of 1-D signals and systems, review of random signals, multi-dimensional signals, review of vector spaces, inner product spaces, orthogonal projections and related concepts.
  • Basics of multi-rate signal processing: sampling, decimation and interpolation, sampling rate conversion (integer and rational sampling rates), oversampled processing (A/D and D/A conversion), and introduction to filter banks.
  • Signal representation: Transform theory and methods (FT and variations, KLT), other transform methods.
  • Wavelets: Characterization of wavelets, wavelet transform, multi-resolution analysis.
  • Statistical signal modeling: The least squares method, Pade’s approximation, Prony’s method, Shanks’ method, iterative pre-filtering, all-pole modeling and linear prediction, autocorrelation and covariance methods, FIR least squares inverse filter design, applications and examples.
  • Inverse problems (signal reconstruction): underdetermined least squares, pseudo-inverse (SVD), min-norm solutions, regularized methods, reconstruction from projections, iterative methods such as projection onto convex sets, expectation-maximization and simulated annealing.

Reference Books:

  • Moon & Stirling, Mathematical Methods and Algorithms for Signal Processing, Prentice Hall, 2000.
  • Monson Hayes, Statistical Digital Signal Processing and Modeling, John Wiley and Sons, 1996.
  • A. Boggess & F. J. Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice Hall, 2001.
  • G. Strang, Introduction to Linear Algebra, 2016.
  • H. Stark & J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 2014.
  • P. P. Vaidyanathan, Multirate systems and filter banks, 1993.

Grading Policy:

  • Homeworks : 25%
  • Mid Term Exams : 25%
  • Project : 25%
  • Final Exam : 25%

Homeworks:

Exams:

Announcements:

Lecture Notes:

 

 

  Topics Covered Lecture Notes
Week 1 Introduction to signal processing; Basics of signals and systems; Linear time-invariant systems; Modes in a linear system; Introduction to state space representation; State space representation; Non-uniqueness of state space representation; Introduction to vector space MMTSP_Week_1
Week 2 Linear independence and spanning set; Unique representation theorem; Basis and cardinality of basis; Norms and inner product spaces; Inner products and induced norm; Cauchy Schwartz inequality; Orthonormality MMTSP_Week_2
Week 3 Linear independence of orthogonal vectors; Hilbert space and linear transformation; Gram Schmidt orthonormalization; Linear approximation of signal space; Gram Schmidt orthogonalization of signals MMTSP_Week_3
Week 4 Basics of probability and random variables; Mean and variance of a random variable; Introduction to random process; Statistical specification of random processes; Stationarity of random processes MMTSP_Week_4
Week 5 Fourier transform of dirac comb sequence; Sampling theorem
Basics of multirate systems; Frequency representation of expanders and decimators; Decimation and interpolation filters		 MMTSP_Week_5
Week 6 Fractional sampling rate alterations; Digital filter banks; DFT as filter bank; Noble Identities; Polyphase representation; Efficient architectures for interpolation and decimation filters MMTSP_Week_6
Week 7 Efficient architecture for fractional decimator; Multistage filter design; Two-channel filter banks; Amplitude and phase distortion in signals MMTSP_Week_7
Week 8 Polyphase representation of 2-channel filter banks, signal flow graphs and perfect reconstruction; M-channel filter banks; Polyphase representation of M-channel filter bank; Perfect reconstruction of signals; Nyquist and half band filters; Special filter banks for perfect reconstruction MMTSP_Week_8
Week 9 Introduction to wavelets; Multiresolution analysis and properties; The Haar wavelet; Structure of subspaces in MRA; Haar decomposition – 1; Haar decomposition – 2 MMTSP_Week_9
Week 10 Wavelet Reconstruction; Haar wavelet and link to filter banks; Demo on wavelet decomposition; Problem on circular convolution; Time frequency localization; Basic analysis: Pointwise and uniform continuity of functions MMTSP_Week_10
Week 11 Basic Analysis : Convergence of sequence of functions; Fourier series and notions of convergence; Convergence of Fourier series at a point of continuity; Convergence of Fourier series for piecewise differentiable periodic functions; Uniform convergence of Fourier series of piecewise smooth periodic function MMTSP_Week_11
Week 12 Convergence in norm of Fourier series; Convergence of Fourier series for all square integrable periodic functions; Matrix Calculus; KL transform; Applications of KL transform; Demo on KL Transform MMTSP_Week_12